You have learned in your general course in quantum mechanics, that the ensemble of the eigenstates of the Hamiltonian of a system forms a complete basis of the space of the states of that system. This remains true for radiation and we are thus going to give the explicit expression of the eigenstates of the Hamiltonian of radiation. We will use what we already know about the space of states of a single mode of radiation So you must be sure to remember the essentials of that previous lesson. We start with the radiation Hamiltonian, which describes a set of independent harmonic oscillators. Two features indicate the independence. First, the total Hamiltonian is a simple sum of individual harmonic oscillator Hamiltonians. Second, the commutators associated with different oscillators are null. We want to solve the eigenequation of H_R. In order to know its eigenvalues and eigenvectors let us explicitly write H_R as a sum of Hamiltonians H_ell. Remember that we know the eigenstates n_ell and eigenvalues E_ell of each individual term H_ell of H_R. The eigenenergies are n_ell + one-half times hbar omega_ell. The corresponding eigenstates called number states are linked to each other by remarkable relations, which I let you contemplate. Do you remember them? There is no option. You must know them. You must also know this remarkable consequence. Any state n_ell can be obtained by applying n_ell times the creation operator to the lowest state, the state with 0 photons in mode ell also called the vacuum of mode ell. To find the eigenvectors of H_R, we can use a remarkably simple mathematical result. States phi that are tensor products of the eigenstate of each H_ell are eigenstates of H_R as you can immediately find by just writing it. The eigenenergy of phi is the sum of eigenenergies of each individual n_ell state. If you are not familiar with tensor products, you can find an excellent presentation of this mathematical tool in the first volume of the course of Cohen-Tannoudji, Diu, and Laloe. But at this stage you can go far by remembering that each operator H_ell only acts upon the states of the mode ell and leaves the states of other modes unchanged. Tensor products are also called outer products. One often uses the simpler notation n_1, n_2, etc, to define such a state. In each mode the index n_ell is a non-negative integer. Such a state is called a Fock state. I already mentioned the name of that Russian pioneer of quantum mechanics. Some people also call such a multimode state, a number state. Coming back to these Fock states, a number state. we can use them as a complete basis of the total radiation space
a number state. we can use them as a complete basis of the total radiation space provided that we consider all the possible values of the integers, n_1 n_2, n_ell, etc. It is a pretty simple, yet powerful result. But there is a price to pay for this simplicity. The dimension of that space is unbelievably large. It scales as the maximum number of photons in each non-empty mode, a number that can be as large as we want, raised to a power that is the number of non-empty modes. Try to imagine such a number. It increases exponentially with the number of non-empty modes. I have a hard time realizing what it means, not you? On the other hand, there is only one state associated with 0 photon in each mode, this is the vacuum. From this unique vacuum we can obtain any Fock state by application of creation operators the right number of times. Here you have an example, where two modes only are not empty. To avoid clumsy notations, one often skips the 0s in such a state and writes it as n_1, n_2, nothing else. All modes not explicitly indicated are implicitly in the vacuum state. In quantum optics, vacuum has intriguing properties which we will discover all along this course. One of them is a fact that the energy of the vacuum is infinite as can be seen when one writes the eigenvalues of the Hamiltonian for any Fock state. This energy is a priori infinite since there is one-half hbar omega_ell term for all the modes, even the empty ones. Physicists have learnt to get around the problem of infinities in quantum radiation and they know how to obtain finite quantities that have an experimental meaning. This is called renormalization. Renormalization is particularly simple in the case of the energy. Let us call E_v, the eigenvalue of H_R in the vacuum, ignoring the fact that it is infinite. We can then define the energy associated with any Fock state, taking E_v as reference. In other words, the energy associated with the state n_1, n2, n_ell is equal to the corresponding eigenvalue of (H_R-E_v). That is the sum over ell of n_ell times hbar omega_ell. This quantity is finite. It is the total energy of n_1 photons, hbar omega_1 plus n_2 photons, hbar omega_2, etc. We have thus been able to get rid of the problem of the infinite value of the energy of the vacuum. This does not mean that we can totally forget that problem. Actually, we will see manifestations of what is called vacuum fluctuations which are related to the one-half terms in the Hamiltonian.